Complex Conjugate
The complex conjugate of a + bi is a - bi, formed by negating the imaginary part while keeping the real part unchanged.
Formula
\overline{a + bi} = a - bi
Definition
The complex conjugate of $a + bi$ is $a - bi$, just flip the sign of the imaginary part, the mirror image of the complex number across the real axis. Key properties: $z\bar{z} = a^2 + b^2 = |z|^2$ is always real and non-negative, $z + \bar{z} = 2a = 2\text{Re}(z)$, and $z - \bar{z} = 2bi$; this lets you divide complex numbers by multiplying numerator and denominator by the conjugate to "rationalize" the denominator. Complex conjugation $z \to \bar{z}$ is the unique non-trivial field automorphism of $\mathbb{C}$ fixing $\mathbb{R}$: it is an involution ($\bar{\bar{z}} = z$) satisfying $\overline{z+w} = \bar{z} + \bar{w}$ and $\overline{zw} = \bar{z}\bar{w}$, and in Hermitian inner product spaces, conjugation appears in the definition $\langle z, w \rangle = \bar{z} w$.
Example
The conjugate of $3 + 5i$ is $3 - 5i$, and the conjugate of $6$ is just $6$ (since $6 = 6 + 0i$). To divide $(2+3i)/(1-2i)$, multiply numerator and denominator by the conjugate $(1+2i)$: $(2+4i+3i+6i^2)/(1+4) = -4/5 + (7/5)i$. The Hermitian transpose (conjugate transpose) of a matrix $A$ is $A^* = \bar{A}^T$, and a matrix is Hermitian if $A^* = A$; all Hermitian matrices have real eigenvalues, generalizing symmetric matrices to complex vector spaces.
Key Insight
Complex roots of polynomials with real coefficients always come in conjugate pairs, if $a + bi$ is a root, so is $a - bi$, which is why quadratic equations have either two real roots or two complex conjugate roots. Conjugation is the automorphism that distinguishes $\mathbb{C}$ from its "mirror image," and the insistence in quantum mechanics that physical observables correspond to Hermitian operators is essentially the requirement that measurement outcomes (eigenvalues) are real, enforced via conjugate symmetry.